Geodesic
Extremum Problem for Periodic Functions Supported in a Ball
Matematičeskie zametki, Tome 69 (2001) no. 3, pp. 346-352

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We consider the Turan $n$-dimensional extremum problem of finding the value of $A_n(hB^n)$ which is equal to the maximum zero Fourier coefficient $\widehat f_0$ of periodic functions $f$ supported in the Euclidean ball $hB^n$ of radius $h$, having nonnegative Fourier coefficients, and satisfying the condition $f(0)=1$. This problem originates from applications to number theory. The case of $A_1([-h,h])$ was studied by S. B. Stechkin. For $A_n(hB^n)$ we obtain an asymptotic series as $h\to0$ whose leading term is found by solving an $n$-dimensional extremum problem for entire functions of exponential type. 
@article{MZM_2001_69_3_a3,
     author = {D. V. Gorbachev},
     title = {Extremum {Problem} for {Periodic} {Functions} {Supported} in a {Ball}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {346--352},
     publisher = {mathdoc},
     volume = {69},
     number = {3},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_69_3_a3/}
}
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D. V. Gorbachev. Extremum Problem for Periodic Functions Supported in a Ball. Matematičeskie zametki, Tome 69 (2001) no. 3, pp. 346-352. http://geodesic.mathdoc.fr/item/MZM_2001_69_3_a3/